Thursday, 27 July 2017

Why mathematics has not been effective in economics

There are three
types of mathematician: those who can count and those who can't.

It takes a few
seconds to get this joke the first time you hear it, and every once
and a while when I tell it to a class one student will raise a hand
and asked for the 'third type'. Its a good joke because it
challenges assumptions, here the assumption is that mathematics is
concerned with arithmetic, which is just a minor branch of number
theory.

At the end of June,
before I went on holiday, I had been thinking about the role of
mathematics in democracy. This was prompted by an invitation from a
Norwegian mathematician to do some work in the topic. I contacted
@BrendanLarvor asking who were the main scholars in the area. He
pointed me to a recent paper that discusses a key issue. Many people
see the role of mathematics in democracy as educating the public so
that they can do their own calculations. Citizens are able to
calculate the cost/benefit of Brexit, for example. But calculation
is not what mathematics is concerned with. The paper is not an easy
read but highlights the awareness amongst (some) mathematicians that
mathematics is not straightforward. In particular the phrase "the
role of mathematics in formatting the world as we experience it"
resonated with me as being a key issue.

This led me to
think about the role of mathematics in defining how the disciplines
of finance and economics are arranged. A consequence of this was I
invited people to answer a short survey on "Does a mathematical
proof enhance a financial theory?". The survey was widely
distributed (via @MarkThoma amongst others) but only elicited seven
responses. The results are here (the survey is still open, btw). I was disappointed that only seven
people seemed to share my interest to the degree that they would
spend a little time answering the question. I concluded that either
people were disinterested (possibly because they thought the question
was trivial, like "Does water flow downhill") or that they
did not understand the question (they do not feel confident about
what is meant by a 'mathematical theory').

On returning from
holiday, I noticed that @freakonometrics had retweeted an article
from aeon about how "By fetishising mathematical models,
economists turned economics into a highly paid pseudoscience"

I have the opinion
that almost all of the criticism of the use of mathematics in
economics stems from a lack of understanding of what mathematics is,
reflecting a general ignorance in economics that has led to the
failure of mathematics in economics. To get an idea of my
frustration consider the following argument about journalism. One
might observe that there are many more photographs in newspapers
today than there were 100 or so years ago. Using the argument that
the problems of economics are in its use of mathematics is rather
like saying the problems of contemporary journalism is down to
photography.

The starting point
of understanding the role of mathematics in finance and economics is
to appreciate what mathematics is concerned with. Mathematics is
concerned with identifying relations between objects: bigger smaller,
to the left/right, symmetry, before/after and so forth. Top class
mathematical research is concerned with discovering new ways of
representing how things are related. More every-day research shows
that A=B or how you go from A to B. Once the mathematicians have
done their work, of "formatting the world as we experience it"
by identifying how we see relations between objects, others then get
on and do things. Mercator figured out how to make maps - a
mathematical operation - sailors then used the maps and in the
process forgot that what they were doing was using mathematics.

Mathematicians rely
on other disciplines providing problems, mathematics, whatever the
caricature of a mathematician dealing with abstract ideals will say.
Mathematics then figures out a way of looking at the problem - the
relations between its components - so that a solution can be found.
The caricature of the mathematician is explained by how mathematics
is presented. Rather than starting with the problem and then
breaking it down into its components, mathematics is presented back
to front. It starts with the components and then shows how these
combine to deliver the observed phenomena. This 'back-to-front'
approach originates in Euclid. The theorems at the end of Euclid's
Elements were all well known hundreds, if not thousands, of years
before he wrote The Elements around 300 BCE.

Euclid's approach is
useful in that it identifies the essential elements of a theorem,
these elements can be the used to construct novel theorems by
combining them in innovative ways; think of a mathematical assumption
as a chemical element and a theorem as a useful molecule. However
there are a number of problems resulting from the way mathematics is
presented. One effect is encapsulated in Kant's argument that
synthetic a priori knowledge was possible. Kant used the example of
Euclid to argue that it was because he had assumed Euclid had deduced
the theorems from first principles. This is significant in that this
fallacious argument was a foundation of Kant's rejection of Hume's
claim that a necessary cause of an effect could never be identified.
Another effect is it provides a model for a powerful rhetorical form
that is persuasive, it was used in particular by Hobbes and Spinoza
while Aquinas' writing has been compared to mathematics. Today
'mathematical' proofs that 1=2 are commonplace. More significantly
this 'mathematical' approach was used by Hobbes to argue that if a
highwayman offered you the choice of 'your money or your life' and
you handed over your money, you were giving consent. It is not easy
to discern flaws in these 'mathematical' arguments, and this is the
day to day job of research mathematicians (a social scientists once
Tweeted they had had a productive day, reviewing three papers: as a
mathematician it will take me a week of hard graft to review a 10
page paper).

The effect in
economics is most clearly seen in Friedman's argument, in the
Methodology of Positive Economics, that the validity of an
economic theorem should not rest on the realism of its assumptions.
I will not dismiss Friedman as the arch-priest of neo-liberalism as I
think the argument he makes has some merits (he focuses on the
empirical outcome and would normally be regarded as 'anti
mathematiciastion'). The attitude he shares with most economists,
along with Kant, Hobbes and Spinoza, is that a 'mathematical'
argument flows from assumptions to conclusions. A mathematician
approach would be to try and tease out the correct assumptions from
the observed behaviour. I would prefer the problem to be re-cast as
"By fetishising synthetic a priori knowledge, economists
turned economics into a highly paid pseudoscience".

The next question is
why do economists do this. The answer is rooted in the observation
that the 'mathematical' approach is powerful rhetorically: you can
use it to convince everyone of almost anything, providing you can
make the chain of arguments tricky enough to follow. From a
philosophical perspective, Kant distinguished the ‘lower
faculties’, such as mathematics, that would consider matters of
pure reason independently of the concerns of the state from the
‘higher faculties’, engineering, jurisprudence, medicine and
theology, were concerned with matters of authority and would be
regulated and monitored by the state. If economics is mathematical
it should inform the state, not be directed by the state, if it is
not then it will have the same status (and funding) as theology (and,
one would suppose, other modern social and human sciences).

More practical
motivations were characterised by Frank Knight, who, around 1920, felt
that economics had split into two strands. There was a mathematical
science, which studied closed systems based on distorting
assumptions, and a descriptive science, which could deduce nothing.
Economics needed to take a middle path that was both realistic and
informative. However, before the Second World War, most economists
doubted the usefulness of mathematics in addressing problems
involving radical uncertainty and human volition, such as the
economy. These attitudes changed when it was seen that mathematics
had transformed how the war, a similarly uncertain and human
activity, had been fought; operations research, cryptography,
supporting the physics of radar and weapons. Based on this
experience and government faith in mathematics, economics began
presenting itself as a mathematical science after the war. Two
publications of 1944 led this transformation: The Probability
Approach in Econometrics by Trygve Håvelmo and The Theory of
Games and Economic Behavior by John von Neumann and Oskar
Morgenstern.

Håvelmo argued also
that if economics wanted to be taken as seriously as physics,
chemistry and biology, it needed to employ probability because that
was the way that opinions were expressed in science. He believed that
if this was done, economics would make new insights, just as
physicists and biologists had. He also observed that the natural
sciences had found a perspective on nature that made it appear to
follow stable laws. The goal of The Probability Approach in
Econometrics was to present how this could be realised.
Morgenstern began The Theory of Games, like Håvelmo, with an
argument for the use of mathematics in economics and explained that
what was required was the careful definition of terms, a
pre-requisite of mathematics but lacking in economics. To this end,
von Neumann started with the axioms of utility that had been at the
core of Carl Menger’s, unmathematical, economics.

When Håvelmo was
awarded the Nobel Prize for Economics in 1989 he reflected that his
aspirations for introducing mathematics to economics had not been
met. He identified the primary issue as being that the economic
models that ‘econometricians’ had been trying to apply to the
data were probably wrong. More fundamentally, economics never
generated new mathematics ‒ ways of seeing relationships ‒ in the
way that the physical sciences had stimulated developments in
mathematics. Economists had simply adapted concepts from other fields
to their own devices.

To my mind, Håvelmo
captures why mathematics is not unreasonable effective in economics.
It is because economists use mathematics as 'part of the plumbing', a
rhetorical tool to convince an audience of an argument. The Unreasonable Effectiveness of Mathematics in the Natural Sciences is
founded on the fact that the natural science use mathematics to
figure out relationships. The one exception to this rule (that I am
aware of) in modern economics is the Fundamental Theorem of Asset
Pricing, formulated by Harrison, Kreps and Pliska around 1980 (I
dismiss game theory as this was originated in the early 1700s). The
FTAP is analogous to the Mercator projection, it describes the basis
on which models (maps) are made that guide probationers (navigators).
“A market admits no arbitrage, if and only if, the market has a martingale measure” establishes
a relationship.

Once mathematics has
delivered ways of identifying relations in physics, 'invariants' can
be identified, such
as momentum, energy or the speed of light (Noether's Theorem is
critical here). Physical theories are then tested on the basis of
whether or not they adhere to a particular conservation law. Because
economics is disinterested in using mathematics to identify
relationships it has been unable to accomplish the next step of
discovering invariants. It has tried, notably by sometimes hoping
'money' is an economic invariant.

In writing Ethics in Quantitative Finance (the points made here are expanded upon in the book) one of my aims was to think of
finance as a mathematician. That is to consider the fundamental relationship, as expressed in the FTAP, and then think
about what this implies as to the fundamental invariant. My
conclusion was that reciprocity - and equality between what is given
and received - is the invariant and I explore why this might be so.
The hope is that finance and economics can actually achieve something
useful for the wider community.

Posted by
Tim Johnson

8 comments:

Why economists have not been effective in economicsComment on Simon Tim Johnson on ‘Why mathematics has not been effective in economics’

Mathematics has not been effective in economics because economics is a cargo cult science. Feynman defined it as follows: “They’re doing everything right. The form is perfect. ... But it doesn’t work. ... So I call these things cargo cult science because they follow all the apparent precepts and forms of scientific investigation, but they’re missing something essential.”

What is missing among economists is a proper understanding of what science is all about. Aristotle gave a working definition 2000+ years ago: “When the premises are certain, true, and primary, and the conclusion formally follows from them, this is demonstration, and produces scientific knowledge of a thing.”

Economists apparently followed this methodology. Walrasian economics is axiomatized, the hard core premises are verbally given as follows: “HC1 economic agents have preferences over outcomes; HC2 agents individually optimize subject to constraints; HC3 agent choice is manifest in interrelated markets; HC4 agents have full relevant knowledge; HC5 observable outcomes are coordinated, and must be discussed with reference to equilibrium states.” (Weintraub)

It should be pretty obvious that the Walrasian axiom set contains THREE NONENTITIES: (i) constrained optimization (HC2), (ii) rational expectations (HC4), (iii) equilibrium (HC5). Every theory/model that contains a nonentity is A PRIORI false. And this is why economics is a cargo cult science. Economists do all the things scientists are supposed to do but it does not work.

Science is about invariances (Nozick) but there is NO such thing as behavioral invariances. Because of this, the Walrasian axioms are methodological madness, to begin with.

Economics suffers from the fact that the subject matter is ill-defined. Economists think that they are doing economics while they bungle amateurishly in sociology and psychology. What economists overlook is that their subject matter is the structure and behavior of the economic system and that all questions about Human Nature/motives/behavior/action are NOT their business.

The task of economics is to figure out how the economy works. Economics is a system science. Accordingly, the correct approach is not microfoundations but macrofoundations.#1

What we have at the moment is Walrasianism, Keynesianism, Marxianism, and Austrianism. Neither of these approaches satisfies the scientific criteria of formal and material consistency. Economists are PROVABLY false with regard to the two most important features of the market economy: (a) the profit mechanism, and (b), the price mechanism. Let this sink in: the profit theory is false since Adam Smith. Instead of having clarified their foundational concepts profit and income, economists have wasted their time fooling around with NONENTITIES.

Economics needs a paradigm shift from false Walrasian microfoundations and false Keynesian macrofoundations to true macrofoundations. Economics is NOT a social science but a system science. A system can be objectively a precisely defined. This is the very condition for the application of mathematics.

When the premises are not correctly defined mathematics cannot work its magic and as a collateral damage econometrics becomes a senseless exercise.#3 When utility maximization is put into the premises no testable proposition ever results. Scientifically incompetent economists do not understand this elementary methodological fact since 140+ years. And this is why mathematics has not been effective in economics.

Hi Egmont, thanks for your thoughts. I'm afraid I don't understand your point about NONENTITIES though. Feel free to spell it out in less technical language...

Isn't the problem just the level of complexity of the system? The fundamental agents in economic models, people, all have huge variation in possible actions. We then have self-consciousness and reflexivity. We react to each other's actions; then, as a system, react again to the changed situation. We further have a kind of collective self-consciousness, which influences the individuals and evolves into a changed state of self-consciousness. This applies to all social sciences. In the case of economics, these complex feedback relationships are particularly strong, in the form of the financial system. The point is that complexity makes the system effectively unpredictable with our current computational power. Even if predictive technology were to improve it would immediately introduce new feedback relationships. All of which means that the equilibrium approach only gives instantaneously valid solutions and is practically useless for prediction.

This may be analogous to your mathematical formulation. In my mind the complexity of the system is economics' fundamental problem. We can do reasonably well using heuristics, so an inductive (historical) approach is preferable to this deductive nonsense, which as you quite rightly point out is based on absurd axioms. This flaw is reasonably uncontroversial in the non-academic economic community, but the majority of academic economists appear to be continuing to behave as if a few lines of equations could predict anything interesting about economic actions.

Samuelson calls Ricardo's theory of comparative advantage the only proposition in the social sciences that is both true and non-trivial. I disagree with the premise - many relationships in macro, such as the Phillips curve, are not quite true, and a mathematician could even (unkindly) call it trivial - but the point is that it's useful. As is Okun's gap. As are the observations underpinning Keynes' general theory. While I can get behind the utility of Walrasian equilibrium, it's been done to death. Further investigations in sub-game perfect Nash equilibria will get us nowhere.

Thanks for your post Egmont, interesting thoughts. I'm afraid I don't understand the point about NONENTITIES though - feel free to expand upon that...

For me the fundamental problem for economics and economists is the complexity of the system. The basic unit of analysis is one person, but you have to predict the whole damn lot of then. They have self-consciousness and reflexivity. The axioms of classical economics, as you rightly point out, are false. But furthermore, they are very, very difficult to predict compared to any unconscious "agent". They even have a level of collective consciousness which makes their actions correspondingly unpredictable. The equilibrium approach is useless where there is this level of dynamism as your solution is only momentarily correct.

What economists ought to do is take a more inductive (historical) approach. This is exactly what the tradition's best thinkers have done, but the approach is not currently popular in academia.

Samuelson's comment about Ricardo's comparative advantage theory being the only proposition in the social sciences that was "both trivial and not obvious" is revealing about what he (and many other orthodox thinkers) thinks economics should do. Why fetishise truth? There is sometimes a trade-off between truth and utility. Some of the most useful economic theories - the Phillips curve, Okun's gap, the observations underpinning Keynes' general theory are not true as such. But they are far more useful in making interesting predictions, to my mind, than extensions of Walrasian equilibrium or sub-game perfect Nash equilibria. This is pretty uncontroversial among the non-academic economic community, but the majority of academics seem not to have substantially adapted their approach. And trivial? Maybe a mathematician could unkindly term macroeconomic theories as such, but this totally misunderstands what economics is trying to do.

Complexity may be a consequence of your more technical formulation of the discipline's problems. But I think it is pretty fundamental to economics' "predictive problem", as we could politely term it.

Great article. As Einstein said: "Math is nature's playbook". See John Cochran "The fact is, we have too little math in economics." Of course it depends on how math is applied or mis-applied. The problem with math in economics is not the math but the economics. You can't run a regression test against my Gospel. It is an ex-ante extrapolation, not ex-post.

Using more sophisticated math, one can easily vastly improve upon my original monetary flows, volume Xs velocity (a surrogate for bank debits, the G.6 release declassified in 1996).

http://monetaryflows.blogspot.com/

See:In 1931 a commission was established on Member Bank Reserve Requirements. The commission completed their recommendations after a 7 year inquiry on Feb. 5, 1938. The study was entitled "Member Bank Reserve Requirements -- Analysis of Committee Proposal"

It's 2nd proposal: "Requirements against debits to deposits"

http://bit.ly/1A9bYH1

After a 45 year hiatus, this research paper was "declassified" on March 23, 1983. By the time this paper was "declassified", Nobel Laureate Dr. Milton Friedman had declared RRs to be a "tax" [sic].

The trajectory for money flows portends a recession (negative growth) in the 2nd qtr. of 2018.

The reason why copper & oil are rising is that long-term flows are rising. The peak is Sept/Oct.

The housing market bust didn't cause the GFC (see Alan Blinder, "After the Music Stopped". Real-estate was too small of a component percentage relative to gDp.

Fascinating article. Falls into the category "Math fails in economics because economists are using math incorrectly." As such, it's a familiar argument. With better application of maths, economists can make it work. Left out of the argument: any evidence for that conclusion...but let's leave that aside. It's a tough problem, after all.

The more fatal weakness of this line of argument seems to me the starting assumptions: mathematics involves "identifying how we see relations between objects." That seems right, but left undefined here is the term "objects." What exactly is an object?

In mathematics, an object is something we can quantify. Now comes the problem: in economics, what we need to identify is the relation between emotions (greed, fear of loss, investor euphoria, etc.) and behavior (buying, selling, tolerance for risk, and so forth).

Alas, this requires that we mathematize emotions. To my knowledge, no one has succeeded in doing this in some 8,000 years of recorded history. Worse: no one has suggested that it is possible, or even meaningful, to attempt quantify & mathematize emotions. Indeed, the general conclusion seems that the effort to do so represents a category error, in the same line as trying to mathematically determine the color of the number two, or the gender of the hypotenuse of a triangle.

Once a mathematician explains to us how to evaluate an expression like "the square root of love divided by the arctangent of fear," we shall surely proceed apace in mathematizing emotions so as to lay economics on a sound mathematical basis.

Until then, this effort seems to belong to the same category of endeavour as applying the Banach-Tarski theorem to a sphere of gold in order to grow rich.

The mathiness problem of economists does not consist in the application of advanced mathematics but in the incapacity to apply the straightforward arithmetic of accounting.

Imagine we have two accountants, one for the business sector, Mr. B, and one for the household sector, Mrs. H. Mr. B is supposed to make an entry every time the firm makes a wage payment and every time the firm sells its output. To make matters simple, the condition of market clearing holds, that is, quantity sold = output, that is, there is no change of inventory. Mrs. H is supposed to make an entry every time one of the households receives wage income and every time a household buys the firm’s product.

Nobody could be more down to earth and historically accurate than Mr. A and Mrs. B. At the end of the first period they meet at the Honest Accountant Bar and compare their numbers, which are shown in the form of accounts on Wikimedia.

Then they calculate their respective balances and find out, to nobody’s, surprise that Qm=-Sm. Note that NO real transactions and transaction entries correspond to the balances. To draw the balances is an ex-post exercise that is NOT backed by a real world transaction.

Next day, the two accountants hand their numbers = Figure (b) over to the economist. Says the economist, hmm, for my purposes I have to rearrange the accounts, after all, profit has to be treated as the income of capital analogous to wage income. I define Gross Domestic Income as GDI ≡ Yw + Qm. He does NOT realize that he puts a flow and a balance together, something no accountant worth his salt would ever do. Now the accounts look like this:

(c) National accounts, consumption expenditures greater than wage income, with profit redefined as a kind of income.https://commons.wikimedia.org/wiki/File:AXEC97.png

The economist now says to himself, obviously, Gross Domestic Income GDI is ‘equal’ to consumption expenditures, which follows from the definitions GDI ≡ Yw + Qm and Qm ≡ C - Yw. Let us call the right-hand side of the business sector’s account Gross Domestic Product GDP for the general case of the sum of consumption expenditures and investment expenditures, i.e. GDP ≡ C + I. . Then we have always GDI ≡ GDP. This, Gross Domestic Income is ‘equal’ to Gross Domestic Product, is the fundamental macroeconomic accounting identity ― the unassailable quantitative/empirical bedrock of economics.

The economists exercise is, of course, futile because profit is NOT the income of capital but the mirror image of dissaving, i.e. the household sector’s increase of debt. Income is a flow and profit is a balance of flows and to lump the two together is sheer stupidity.

From the graphics it is immediately obvious that Keynes’ foundational identity “Income = value of output” is false. Why? Because Keynes did not come to grips with profit: “His Collected Writings show that he wrestled to solve the Profit Puzzle up till the semi-final versions of his GT but in the end he gave up and discarded the draft chapter dealing with it.” (Tómasson et al.)#1

Because economists ― Keynes, Keynesians, Post Keynesians, Anti-Keynesians and all the rest ― cannot even do the elementary mathematics of accounting the profit theory is false since Adam Smith.#2 This means, because economists cannot do their little math ALL of economics is proto-scientific rubbish.

Egmont Kakarot-Handtke

#1 See also ‘Economists do not solve problems, they are the problem’https://axecorg.blogspot.de/2015/01/economists-do-not-solve-problems-they.html

#2 For more details see cross-references Accountinghttp://axecorg.blogspot.de/2016/12/accounting-cross-references.html